Explanation Formula used: The governing equation of motion is m d 2 y d t 2 + δ d y d t + k y = 0 . The Hooks Law is as F = k x . The quadratic equation a x 2 + b x + c = 0 is calculated as x = − b ± b 2 − 4 a c 2 a . Theorem used: If r 1 = α + j β and r 1 = α − j β are the two complex roots of the auxiliary equation a r 2 + b r + c = 0 , then y = e α x ( c 1 cos β x + c 2 sin β x ) is the general solution of a y ″ + b y ′ + c y = 0 . Calculation: The eight of the spring is 4 .00 lb with initial stretch( x ) is 0.125 ft, the spring is pulled by 3.0 in longer than the equilibrium length and then released, and the damping force is equal to the velocity. From the Hooks Law, F = k x . Where k is spring constant, x is displacement of spring and F is force applied on the spring. Substitute the value of F and x in F = k x . F = k x 4.00 = k × 0.125 k = 4 0.125 k = 32 lb/ft Now, to get the mass of object: F = m a 4.00 = m × 32 m = 4 32 = 0.125 slugs So, the governing equation of motion is as follows: m d 2 y d t 2 + δ d y d t + k y = 0 . Substitute the value of m , δ and k in m d 2 y d t 2 + δ d y d t + k y = 0 . 0.125 d 2 x d t 2 + 1 × d x d t + 4 x = 0 d 2 x d t 2 + 8 d x d t + 32 x = 0 The auxiliary equation is: m 2 + 8 m + 32 = 0 Compare the above equation with a x 2 + b x + c = 0 . Here a = 1 , b = 8 , c = 32 . Substitute the value of a = 1 , b = 8 , c = 32 in the quadratic formula x = − b ± b 2 − 4 a c 2 a . m = − 8 ± 8 2 − 4 × 1 × 32 2 × 1 = − 8 ± 64 − 128 2 = − 4 ± − 16 = − 4 ± 4 j So, the general solution to the above differential equation is x = e − 4 t ( c 1 sin 4 t + c 2 cos 4 t ) Substitute the initial conditions x = 0

Basic Technical Mathematics with Calculus (11th Edition)

11th Edition

ISBN: 9780134437736

Author: Allyn J. Washington, Richard Evans

Publisher: PEARSON

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Chapter 31.10, Problem 14E

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The equation of resulting motion if a damping force numerically equal to the velocity is present.

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Chapter 31.10, Problem 13E

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Chapter 31 Solutions

Basic Technical Mathematics with Calculus (11th Edition)

Ch. 31.1 - Show that is a solution of . Is it the general...Ch. 31.1 - Prob. 1E Ch. 31.1 - In Exercises 1 and 2, show that the indicated...Ch. 31.1 - In Exercises 3–6, determine whether the given...Ch. 31.1 - Prob. 4E Ch. 31.1 - In Exercises 3–6, determine whether the given...Ch. 31.1 - Prob. 6E Ch. 31.1 - In Exercises 7–10, show that each function y =...Ch. 31.1 - Prob. 8E Ch. 31.1 - In Exercises 7–10, show that each function y =...

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